#9498966

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#55107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,14,17)(11,15,20,23)(12,13,21,16)(18,19,24,22)(25,26)(27,28)(29,34)(30,35)(31,32)(33,40)(36,38)(37,39);(1,2)(3,4)(5,7)(6,8)(9,11,14,20)(10,15,17,23)(12,18,21,24)(13,19,16,22)(26,32)(27,33)(34,35)(36,37);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,17,13)(11,18,20,24)(15,22,23,19)(25,27,31,33)(26,28,32,40)(29,36,30,37)(34,38,35,39);(1,4,5,6)(2,3,7,8)(9,12)(10,13)(11,18)(14,21)(15,19)(16,17)(20,24)(22,23)(25,28)(26,33)(27,32)(29,38)(30,39)(31,40)(34,37)(35,36);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29,31,30)(26,34,32,35)(27,36,33,37)(28,38,40,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,12)(11,19)(14,16)(15,18)(17,21)(20,22)(23,24)(25,30,31,29)(26,35,32,34)(27,37,33,36)(28,39,40,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9406162);